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Asia-Pacific Journal of Operational ResearchVol. 22, No. 1 (2005) 19–31c© World Scientific Publishing Co. & Operational Research Society of Singapore

NONDIFFERENTIABLE SECOND-ORDERSYMMETRIC DUALITY

I. AHMAD∗ and Z. HUSAIN

Department of Mathematics, Aligarh Muslim UniversityAligarh -202 002, India∗iahmad@postmark.net

Received June 2003Revised February 2004

A pair of Mond–Weir type nondifferentiable second-order symmetric primal and dualproblems in mathematical programming is formulated. Weak duality, strong duality, andconverse duality theorems are established under η-pseudobonvexity assumptions. Sym-metric minimax mixed integer primal and dual problems are also investigated. Moreover,the self duality theorem is also discussed.

Keywords: Second-order symmetric duality; nondifferentiable programming; minimax;integer programming; self-duality.

1. Introduction

Dantzig et al. (1965) first formulated a pair of the symmetric dual nonlinearprograms in which the dual of the dual equals the primal and established weakand strong duality for these problems concerning convex and concave functions.Chandra and Husain (1981) studied a pair of symmetric dual nondifferentiableprograms assuming convexity–concavity assumptions. Subsequently, Chandra et al.(1985) presented another pair of symmetric dual nondifferentiable programs in thespirit of Mond and Weir (1981), and discussed duality involving pseudoconvexity–pseudoconcavity. Recently, Lalitha et al. (2003) derived duality theorems for mul-tiobjective fractional symmetric dual programs under invexity.

Mangasarian (1975) considered a nonlinear program and discussed second-orderduality under certain inequalities. Mond (1974) assumed rather simple inequali-ties. Later on, Bector and Chandra (1986) defined the functions satisfying theseinequalities to be bonvex–boncave. To give examples of bonvex–boncave functions,Mond (1974) has shown that a convex (concave) quadratic or linear function is bon-vex (boncave). Mangasarian (1975, p. 609), Mond (1974, p. 93) and Hanson (1993,p. 316) have indicated possible computational advantages of second-order dual overthe first-order dual.

19

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20 I. Ahmad & Z. Husain

Pandey (1965) introduced the concept of η-bonvex functions as an extensionof bonvex functions. Mishra (2000) studied second-order symmetric duality undersecond-order F-convexity and second-order F-pseudoconvexity for Wolfe and Mond–Weir type models. Recently, Yang et al. (2003) formulated a pair of Wolfe typenondifferentiable second-order symmetric nonlinear programs involving a supportfunction in the objective function and discussed the duality results under second-order F-convexity assumptions.

In the present study, we construct a new pair of Mond–Weir type nondiffer-entiable second-order symmetric dual programs and establish weak duality, strongduality, and converse duality theorems under the assumptions of η-pseudobonvexity.These second-order duality results are then used to investigate symmetric dualityfor a minimax version of second-order symmetric dual programs wherein some ofthe primal and dual variables are constrained to belong to some arbitrary sets, e.g.the sets of integers. Finally, the self-duality theorem is also discussed.

2. Notations and Preliminaries

Let Rn be the n-dimensional Euclidean space and K(x, y): X×Y → R be a real val-ued twice continuously differentiable function defined on an open set in Rn+m. LetKx(x, y) denotes the gradient vector of K with respect to x at (x, y) and Kxx(x, y)the Hessian matrix with respect to x evaluated at (x, y). Ky(x, y), Kxy(x, y), andKyy(x, y) are defined similarly.

Definition 2.1 A real valued twice differentiable function K(x, y): X × Y → R,where X and Y are open sets in Rn and Rm, respectively, is said to beη-pseudobonvex in the first variable at u ∈ X for fixed v ∈ Y , if there exists afunction η: X × X → Rn such that for r ∈ Rn, x ∈ X,

ηt(x, u)[Ku(u, v) + Kuu(u, v)r] ≥ 0 ⇒ K(x, v) ≥ K(u, v) − 12rtKuu(u, v)r.

A real valued twice differentiable function K(x, y): X × Y → R is said to beη-pseudoboncave if −K is η-pseudobonvex.

In what follows, we require the following notion of separability of a vector func-tion (Balas, 1970; Gulati and Ahmad, 1997).

Definition 2.2 Let s1, s2, . . . , sr be the elements of an arbitrary vector space. Avector function L(s1, s2, . . . , sr) will be called additively separable with respectto s1 if there exist vector functions M(s1) (independent of s2, s3, . . . , sr) andN(s2, s3, . . . , sr) (independent of s1) such that

L(s1, s2, . . . , sr) = M(s1) + N(s2, s3, . . . , sr).

3. Mond–Weir Type Symmetric Duality

We now present the following pair of second-order symmetric nondifferentiable non-linear programs and establish weak, strong, and converse duality theorems.

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Nondifferentiable Second-Order Symmetric Duality 21

Primal (MP):

Minimize G(x, y, w, p) = K(x, y) + (xtBx)12 − ytCw − 1

2ptKyy(x, y)p

subject to

Ky(x, y) − Cw + Kyy(x, y)p ≤ 0, (3.1)

yt{Ky(x, y) − Cw + Kyy(x, y)p} ≥ 0, (3.2)

wtCw ≤ 1, (3.3)

x ≥ 0. (3.4)

Dual (MD):

Maximize H(u, v, z, r) = K(u, v) − (vtCv)12 + utBz − 1

2rtKuu(u, v)r

subject to

Ku(u, v) + Bz + Kuu(u, v)r ≥ 0, (3.5)

ut{Ku(u, v) + Bz + Kuu(u, v)r} ≤ 0, (3.6)

ztBz ≤ 1, (3.7)

v ≥ 0, (3.8)

where

(i) p, w are m dimensional and r, z, are n-dimensional vectors, respectively; and(ii) B and C are positive semidefinite matrices of order n and m, respectively.

Lemma 3.1 (Generalized Schwartz Inequality). Let A be a positive semidefinitesymmetric matrix of order n. Then, for all x, z ∈ Rn,

xtAz ≤ (xtAx)12 (ztAz)

12 .

Theorem 3.1 (Weak Duality). Let (x, y, w, p) be feasible for the primal problem(MP) and (u, v, z, r) be feasible for the dual problem (MD). Suppose that there existfunctions η: Rn × Rn → Rn and ξ: Rm × Rm → Rm satisfying

(i) η(x, u) + u ≥ 0, and(ii) ξ(v, y) + y ≥ 0.

Furthermore, assume that K(·, v) + (·)tBz is η-pseudobonvex at u and K(x, ·) −(·)tCw is ξ-pseudoboncave at y. Then,

Inf(MP ) ≥ Sup(MD).

Proof. Equation (3.5) along with hypothesis (i) implies

ηt(x, u)[Ku(u, v) + Bz + Kuu(u, v)r]

≥ −ut[Ku(u, v) + Bz + Kuu(u, v)r] ≥ 0 (by (3.6)).

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22 I. Ahmad & Z. Husain

Using η-pseudobonvexity of K(·, v) + (·)tBz, we have

K(x, v) + xtBz ≥ K(u, v) + utBz − 12rtKuu(u, v)r. (3.9)

By (3.1) and hypothesis (ii), we get

ξt(v, y)[Ky(x, y) − Cw + Kyy(x, y)p]

≤ −yt[Ky(x, y) − Cw + Kyy(x, y)p] ≤ 0 (from (3.2)),

which by ξ-pseudoboncavity of K(x, ·) − (·)tCw yields

K(x, v) − vtCw ≤ K(x, y) − ytCw − 12ptKyy(x, y)p. (3.10)

Combining inequalities (3.9) and (3.10), we obtain

K(x, y) + xtBz − ytCw − 12ptKyy(x, y)p

≥ K(u, v) + utBz − vtCw − 12rtKuu(u, v)r,

which along with Lemma 3.1 and Eqs. (3.3) and (3.7) gives

K(x, y) + (xtBx)12 − ytCw − 1

2ptKyy(x, y)p

≥ K(u, v) − (vtCv)12 + utBz − 1

2rtKuu(u, v)r,

and hence

Inf(MP ) ≥ Sup(MD).

Theorem 3.2 (Strong Duality). Let K: Rn ×Rm → R be thrice differentiable andlet (x, y, w, p) be a local optimal solution of the primal problem (MP). If

(a) Kyy(x, y) is nonsingular,(b) the matrix (Kyy(x, y)p)y is positive or negative definite, and(c) Ky(x, y) − Cw + Kyy(x, y)p �= 0,

then, there exists z ∈ Rn such that p = 0, (x, y, z, r = 0) is feasible for the dualproblem (MD), and the two objectives have the same values.

Also, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions ofthe primal and dual problems, then (x, y, w, p) and (x, y, z, r) are global optimalsolutions for (MP) and (MD), respectively.

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Nondifferentiable Second-Order Symmetric Duality 23

Proof. Since (x, y, w, p) is a local optimal solution of (MP), by the Fritz–Johnoptimality conditions (Craven and Mond, 1976), there exist α ∈ R, β ∈ Rm, γ ∈ R,ν ∈ R, and δ ∈ Rn such that

α[Kx + Bz − 1

2 (Kyyp)xp]+ [Kyx + (Kyy p)x] (β − γy) − δ = 0, (3.11)

α[Ky − Cw − 1

2 (Kyy p)y p]+ [Kyy + (Kyy p)y] (β − γy)

− γ [Ky − Cw + Kyyp] = 0, (3.12)

αCy + (β − γy)tC = 2νCw, (3.13)

(β − γy − αp)tKyy = 0, (3.14)

βt [Ky − Cw + Kyyp] = 0, (3.15)

γyt [Ky − Cw + Kyyp] = 0, (3.16)

ν(wtCw − 1) = 0, (3.17)

xtδ = 0, (3.18)

ztBz ≤ 1, (3.19)

xtBz = (xtBx)12 , (3.20)

(α, β, γ, ν, δ) ≥ 0, (3.21)

(α, β, γ, ν, δ) �= 0. (3.22)

Since Kyy is nonsingular, (3.14) yields

β = αp + γy. (3.23)

Therefore (3.11) and (3.12) give

α[Kx + Bz − 1

2 (Kyyp)xp]+ [Kyx + (Kyyp)x] αp − δ = 0 (3.24)

and

(α − γ) [Ky − Cw + Kyyp] + 12 (Kyy p)y(β − γy) = 0. (3.25)

On multiplying (3.25) by (β−γy)t from the left and using (3.15) and (3.16), we get

(β − γy)t(Kyy p)y(β − γy) = 0, (3.26)

which by the hypothesis (b) implies

β = γy. (3.27)

Therefore, from (3.25)

(α − γ)(Ky − Cw + Kyyp) = 0, (3.28)

which by the hypothesis (c) yields

α = γ. (3.29)

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24 I. Ahmad & Z. Husain

Suppose α = γ = 0. Then, from (3.24) and (3.27), we obtain δ = 0 and β = 0,

respectively. Also, Eq. (3.13) gives ν = 0 if Cw �= 0. Thus, (α, β, γ, ν, δ) = 0, acontradiction to (3.22). Hence

γ = α > 0. (3.30)

Now from (3.23), (3.27), and (3.30)

p = 0 (3.31)

and

y =β

γ≥ 0. (3.32)

Moreover, (3.31) along with (3.24) and (3.18) yields

[Kx + Bz] =δ

α≥ 0

and

xt [Kx + Bz] =xtδ

α= 0.

Therefore, (x, y, z, r = 0) is feasible for (MD).Now let 2ν

α = a. Then, a ≥ 0, and from (3.13) and (3.27),

Cy = aCw, (3.33)

which is the condition for equality in Lemma 3.1. Therefore,

ytCw = (ytCy)12 (wtCw)

12 .

In the case ν > 0, (3.17) gives wtCw = 1 and so ytCw = (ytCy)12 . In the case,

ν = 0, (3.33) gives Cy = 0 and so ytCw = (ytCy)12 = 0. Thus, in either case,

ytCw = (ytCy)12 . (3.34)

Hence

G(x, y, w, p = 0) = K(x, y) + (xtBx)12 − ytCw

= K(x, y) − (ytCy)12 + xtBz (using (3.20) and (3.34))

= H(x, y, z, r = 0).

Also, by Theorem 3.1, (x, y, w, p) and (x, y, z, r) are global optimal solutions for(MP) and (MD), respectively.

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Nondifferentiable Second-Order Symmetric Duality 25

Theorem 3.3 (Converse Duality). Let K: Rn × Rm → R be thrice differentiableand let (u, v, z, r) be a local optimal solution of the dual problem (MD). If

(a) Kuu(u, v) is nonsingular,(b) the matrix (Kuu(u, v)r)u is positive or negative definite, and(c) Ku(u, v) + Bz + Kuu(u, v)r �= 0,

then there exists w ∈ Rm such that r = 0, (u, v, w, p = 0) is feasible for the primalproblem (MP), and the two objectives have the same values.

Also, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions ofthe primal and dual problems, then (u, v, w, p) and (u, v, z, r) are global optimalsolutions for (MP) and (MD), respectively.

Proof. It follows on the lines of Theorem 3.2.

4. Minimax Mixed Integer Symmetric Programs

Let U and V be two arbitrary sets of integers in Rn and Rm respectively. In thepresent section, we constrain some of the components of x and y to belong toarbitrary sets of integers as in Balas (1970) and Gulati and Ahmad (1997). Supposethe first n1 (0 ≤ n1 ≤ n) components of x belong to U and the first m1 (0 ≤m1 ≤ m) components of y belong to V . Then, we write [x, y] = [x1, x2, y1, y2],where x1 = [x1, x2, . . . , xn1 ] and y1 = [y1, y2, . . . , ym1 ], x2 and y2 being the vectorsof the remaining components of x and y, respectively.

For a twice differentiable function K in the components of x2 and y2, letKx2(x, y) and Ky2(x, y) be the partial derivatives of K with respect to x2 and y2,respectively, evaluated at (x, y). Also, let Kx2x2(x, y) denote the Hessian matrixwith respect to x2 evaluated at (x, y). Ky2y2(x, y) and Kx2y2(x, y) are definedsimilarly.

We now formulate the maxmin version of a pair of dual programs considered inSection 3.

Primal (SP):

Maxx1 Minx2,y,w K(x, y) + ((x2)tBx2)12 − (y2)tCw − 1

2ptKy2y2(x, y)p

subject to

Ky2(x, y) − Cw + Ky2y2(x, y)p ≤ 0,

(y2)t{Ky2(x, y) − Cw + Ky2y2(x, y)p

} ≥ 0,

wtCw ≤ 1,

x2 ≥ 0,

x1 ∈ U, y1 ∈ V.

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26 I. Ahmad & Z. Husain

Dual (SD):

Minv1 Maxu,v2,z K(u, v) − ((v2)tCv2)12 + (u2)tBz − 1

2rtKu2u2(u, v)r

subject to

Ku2(u, v) + Bz + Ku2u2(u, v)r ≥ 0,

(u2)t {Ku2(u, v) + Bz + Ku2u2(u, v)r} ≤ 0,

ztBz ≤ 1,

v2 ≥ 0,

u1 ∈ U, v1 ∈ V,

where

(i) p, w are m−m1 dimensional and r, z are n−n1 dimensional vectors, respec-tively; and

(ii) B and C are positive semidefinite matrices of order n − n1 and m − m1,respectively.

Theorem 4.1 (Symmetric Duality). Let (x, y, w, p) be an optimal solution for (SP).Suppose that the following conditions are satisfied:

(i) K(x, y) is additively separable with respect to x1 or y1,

(ii) for any feasible solution (x, y, w, p) in (SP) and any feasible solution (u, v, z, r)in (SD), K(u, v)+ (u2)tBz is η-pseudobonvex in u2 with respect to r ∈ Rn−n1

for each (u1, v, z) and K(x, y)−(y2)tCw is ξ-pseudoboncave in y2 with respectto p ∈ Rm−m1 for each (x, y1, w),

(iii) K(x, y) is thrice differentiable in x2 and y2,

(iv) the matrix (Ky2y2(x, y)p)y2 is positive or negative definite,(v) Ky2(x, y) − Cw + Ky2y2(x, y)p �= 0,

(vi) η(x2, u2) + u2 ≥ 0, and(vii) ξ(v2, y2) + y2 ≥ 0.

Then, there exists z ∈ Rn−n1 such that p = 0, (x, y, z, r = 0) is optimal for (SD)and the two objectives have the same values.

Proof. Let

q = Maxx1 Minx2,y,w

× [K(x, y) + ((x2)tBx2)12 − (y2)

tCw − 1

2ptKy2y2(x, y)p : (x, y, w, p) ∈ Q]

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Nondifferentiable Second-Order Symmetric Duality 27

and

s = Minv1 Maxu,v2,z

× [K(u, v) − ((v2)tCv2)12 + (u2)

tBz − 1

2rtKu2u2(u, v)r : (u, v, z, r) ∈ S],

where Q and S are feasible regions of primal (SP) and dual (SD), respectively.As K(x, y) is taken to be additively separable with respect to x1 or y1 (say with

respect to x1), it follows that

K(x, y) = K1(x1) + K2(x2, y).

Therefore, Ky2(x, y) = K2y2(x2, y), Ky2y2(x, y) = K2

y2y2(x2, y) and q can bewritten as

q = Maxx1 Minx2,y,w

×[K1(x1) + K2(x2, y) + ((x2)tBx2)

12 − (y2)tCw − 1

2ptK2y2y2(x2, y)p :

K2y2(x2, y) − Cw + K2

y2y2(x2, y)p ≤ 0,

(y2)t{K2y2(x2, y) − Cw + K2

y2y2(x2, y)p} ≥ 0,

wtCw ≤ 1,

x2 ≥ 0, x1 ∈ U, y1 ∈ V].

Or

q = Maxx1 Miny1 [K1(x1) + φ(y1) : x1 ∈ U, y1 ∈ V ],

where

φ(y1) = Minx2,y2,w

×[K2(x2, y) + ((x2)tBx2)

12 − (y2)tCw − 1

2ptK2y2y2(x2, y)p :

K2y2(x2, y) − Cw + K2

y2y2(x2, y)p ≤ 0,

(y2)t{K2y2(x2, y) − Cw + K2

y2y2(x2, y)p} ≥ 0,

wtCw ≤ 1,

x2 ≥ 0, x1 ∈ U, y1 ∈ V]. (4.1)

Similarly,

s = Minv1 Maxu1 [K1(u1) + ψ(v1) : u1 ∈ U, v1 ∈ V ],

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28 I. Ahmad & Z. Husain

where

ψ(v1) = Maxu2,v2,z

[K2(u2, v) − ((v2)tCv2)

12 + (u2)tBz − 1

2rtK2u2u2(u2, v)r :

K2u2(u2, v) + Bz + K2

u2u2(u2, v)r ≥ 0,

(u2)t{K2u2(u2, v) + Bz + K2

u2u2(u2, v)r} ≤ 0,

ztBz ≤ 1,

v2 ≥ 0, u1 ∈ U, v1 ∈ V]. (4.2)

For any given y1, programs (4.1) and (4.2) are a pair of Mond–Weir type nondif-ferentiable second-order symmetric dual programs given in Section 3 and hence, inview of the various hypotheses made here, Theorems 3.1 and 3.2 become applicable.Therefore, for y1 = y1, we have

φ(y1) = ψ(y1).

It remains to prove now that (x, y, z, r = 0) solves the problem (SD). Theproof of this part follows on the lines of Gulati and Ahmad (1997) and Gulatiet al. (1997).

5. Self-Duality

A mathematical programming problem is said to be self-dual if it is formally iden-tical with its dual, that is, the dual can be recast in the form of the primal. If weassume

(i) K to be skew symmetric, that is, K(x, y) = −K(y, x), and(ii) B = C,

then, we shall show that the programs (SP) and (SD) are self-duals. By recastingthe dual problem (SD) as a maxmin problem, we have

Maxv1 Minu,v2,z − K(u, v) + ((v2)tCv2)12 − (u2)

tBz + 1

2rtKu2u2(u, v)r

subject to

−Ku2(u, v) − Bz − Ku2u2(u, v)r ≤ 0,

(u2)t[−Ku2(u, v) − Bz − Ku2u2(u, v)r] ≥ 0,

ztBz ≤ 1,

v2 ≥ 0,

u1 ∈ U, v1 ∈ V.

Since K is skew symmetric

Ku2(u, v) = −Ku2(v, u), Ku2u2(u, v) = −Ku2u2(v, u).

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Nondifferentiable Second-Order Symmetric Duality 29

Also, since B = C, the above problem becomes

Maxv1 Minv2,u,z K(v, u) + ((v2)tBv2)12 − (u2)

tCz − 1

2rtKu2u2(v, u)r

subject to

Ku2(v, u) − Cz + Ku2u2(v, u)r ≤ 0,

(u2)t[Ku2(v, u) − Cz + Ku2u2(v, u)r] ≥ 0,

ztCz ≤ 1,

v2 ≥ 0,

u1 ∈ U, v1 ∈ V,

which is the primal problem (SP). Hence (SP) is a self-dual.We now state the self-duality theorem. Its proof follows on the lines of Chandra

and Husain (1981) and Gulati and Ahmad (1997).

Theorem 5.1 (Self-Duality). Let K: Rn × Rn → R be skew symmetric andB =C. Then (SP) is a self-dual. Furthermore, if (SP) and (SD) are dual pro-grams and (x, y, w, p) is an optimal solution for (SP), then (x, y, w, p = 0) and(y, x, z, r = 0) are optimal solutions for (SP) and (SD) respectively, and

Maxmin (SP ) = 0 = Minmax (SD).

6. Special Cases

(i) Let U = φ, V = φ. Then (SP) and (SD) are reduced to the second-ordersymmetric dual programs of Section 3. Moreover, if p = 0 and r = 0, then(SP) and (SD) become the symmetric dual programs of Chandra et al. (1985).

(ii) Let B = 0 and C = 0. Then (SP) and (SD) become the second-order symmetricminimax mixed integer dual programs of Gulati and Ahmad (1997). If, inaddition, U = φ and V = φ, then, we get the second-order symmetric dualprograms of Gulati et al. (2001).

(iii) If in the problems (SP) and (SD), the second-order terms are omitted, i.e.,p and r are set to zero vectors, then they become the first-order nondiffer-entiable symmetric dual mixed integer programs of Gulati et al. (1997). Ifin addition, U = φ, V = φ, B = 0, and C = 0, then we get the first-ordersymmetric dual programs of Mond and Weir (1981).

7. Conclusion

In this article, we presented duality relations between second-order nondifferen-tiable symmetric primal and dual problems. These results were generalized to min-imax mixed integer analogue wherein some of the primal and dual variables areconstrained to belong to some arbitrary sets, e.g., the set of integers. The resultsdeveloped in this paper improve and generalize a number of existing results in the

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30 I. Ahmad & Z. Husain

literature. Using global invexity assumptions the results of this paper can be furthergeneralized along the lines of Craven (2002).

Acknowledgments

The authors wish to thank the referees for their valuable suggestions which haveimproved the presentation of this paper.

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I. Ahmad is a Lecturer in the Department of Mathematics, Aligarh MuslimUniversity, Aligarh, India. He obtained his M.Sc. in Applied Mathematics andM.Phil. and Ph.D. in Mathematics from University of Roorkee, India (presentlyI.I.T. Roorkee, India). He has published a number of papers in the area of mathe-matical programming.

Z. Husain is a Research Scholar in the Department of Mathematics, AligarhMuslim University, Aligarh, India. He obtained his M.Sc. from M.J.P. RohilkhandUniversity, Bareilly, India and M.Phil. from Aligarh Muslim University, Aligarh,India.